Electronic edition published by Cultural Heritage Langauge Technologies (with permission from Charles Scribners and Sons) and funded by the National Science Foundation International Digital Libraries Program. This text has been proofread to a low degree of accuracy. It was converted to electronic form using data entry.
NEWTON, ISAAC (b. Woolsthorpe, England,
25 December 1642; d. London, England, 20 March
1727), mathematics, dynamics, celestial mechanics,
astronomy, optics, natural philosophy.
Mathematics. Any summary of Newton's contributions
to mathematics must take account not only
of his fundamental work in the calculus and other
aspects of analysis--including infinite series (and most
notably the general binomial expansion)--but also his
activity in algebra and number theory, classical and
analytic geometry, finite differences, the classification
of curves, methods of computation and approximation,
and even probability.
At the beginning of my mathematical studies, when I
had met with the works of our celebrated Wallis, on
considering the series, by the intercalation of which he
himself exhibits the area of the circle and the hyperbola,
the fact that in the series of curves whose common base
or axis is x and the ordinates
x, x - 1/3x3, x - 2/3x3 + 1/5x5, x -
3/3x3 + 3/5x5 - 1/7x7, etc.
could be interpolated, we would have the areas of the
intermediate ones, of which the first (1 -
x2)1/2 is the
circle....26
The importance of changing Wallis' fixed upper
boundary to a free variable x has been called “the crux
of Newton's breakthrough,” since the “various
powers of x order the numerical coefficients and reveal
for the first time the binomial character of the
sequence.”27
In about 1665, Newton found the power series
(that is, actually determined the sequence of the
coefficients) for
sin-1x = x + 1/6x3 + 3/40x5 +
...,
and--most important of all--the logarithmic series.
He also squared the hyperbola y(1 + x) = 1, by
tabulating
0?x (1 + t)r dt
for r = 0, 1, 2, ... in powers of x and then
interpolating
0?x (1 + t)-1
dt.28
From his table, he found the square of the hyperbola
in the series
which is the series for the natural logarithm of 1 + x.
Newton wrote that having “found the method of
infinite series,” in the winter of 1664-1665, “in summer
1665 being forced from Cambridge by the Plague
I computed the area of the Hyperbola at Boothby ...
to two & fifty figures by the same method.”29
At about the same time Newton devised “a completely
general differentiation procedure founded on
the concept of an indefinitely small and ultimately
vanishing element o of a variable, say, x.” He first
used the notation of a “little zero” in September 1664,
in notes based on Descartes's Géométrie, then extended
it to various kinds of mathematical investigations.
From the derivative of an algebraic function f(x)
conceived (“essentially”) as
Lim.(0?zero)1/0 [f(x + 0) - f(x)]
he developed general rules of differentiation.
The next year, in Lincolnshire and separated from
books, Newton developed a new theoretical basis for
his techniques of the calculus. Whiteside has summarized
this stage as follows:
[Newton rejected] as his foundation the concept of
the indefinitely small, discrete increment in favor of
that of the “fluxion” of a variable, a finite instantaneous
speed defined with respect to an independent, conventional
dimension of time and on the geometrical model
of the line-segment: in modern language, the fluxion of
the variable x with regard to independent time-variable t
is the “speed” dx/dt.30
Prior to 1691, when he introduced the more familiar
dot notation (? for dx/dt, ? for
dy/dt, ? for dz/dt; then
? for d2x/dt2, ? for
d2y/dt2, ? for
d2z/dt2), Newton
generally used the letters p, q, r for the first
derivatives
(Leibnizian dx/dt, dy/dt, dz/dt) of variable quantities
x, y, z, with respect to some independent variable
t.
In this scheme, the “little zero” o was “an
arbitrary increment of time,”31 and op, oq,
or were
the corresponding “moments,” or increments of the
variables x, y, z (later these would, of course, become
o?, o?, o?).32 Hence, in
the limit (o ? zero), in the
modern Leibnizian terminology
q/p = dy/dx r/p = dz/dx,
where “we may think of the increment o as absorbed
into the limit ratios.” When, as was often done for the
sake of simplicity, x itself was taken for the independent
time variable, since x = t, then p = ? =
dx/dx = 1, q = dy/dx, and r = dz/dx.
In May 1665, Newton invented a “true partial-derivative
symbolism,” and he “widely used the
notation ? and ? for the respective homogenized
derivatives x(dp/dx) and
x2(d2p/dx2),” in particular
to express the total derivative of the function
?i (piyi) = 0
before “breaking through ... to the first recorded use
of a true partial-derivative symbolism.” Armed with
this tool, he constructed “the five first and second
order partial derivatives of a two-valued function”
and composed the fluxional tract of October 1666.33
